(This is one of a series of GMAT tips that we offer on our blog.)

The quantitative section of the GMAT has a known emphasis on factorization of numbers, asking a variety of questions about divisibility, primes, least common multiple, etc. One fairly common question type asks, “How many unique factors does (number) have?”

While there are certainly strategies to answer this question efficiently without any kind of shortcut trick, a pretty slick shortcut does exist:

The first step would likely be your first step in any factorization problem; break the number down in to its prime factors. Take a number like 24, which breaks in to the factors 2*2*2*3.

Secondly, express that factorization as the product of exponents. In this case, it’s 2^3 * 3^1.

Next, discard the bases, and add one to each of the exponents. Here we’d have 3 and 1, and add one to each to make them 4 and 2.

Then, multiply the exponents-plus-one. 4*2 is 8, and there are 8 unique factors of 24:

1, 2, 3, 4, 6, 8, 12, and 24

The proof is a bit messy, but derives from another GMAT concept — combinatorics — which relies heavily on the use of, of all things, factorials.

If you blank on a trick like this, however, note that you can systematically come up with the above list of factors on your own, by taking the prime factors (2, 2, 2, and 3) and 1 (a factor of any integer), and multiplying each possible combination of them:

1

1*2 = 2

1 * 3 = 3

2*2 = 4

2 * 2 * 2 = 8

2 * 3 = 6

2 * 2 * 3 = 12

2 * 2 * 2 * 3 = 24

It’s a bit more time consuming, but doesn’t require that you memorize a trick precisely in order to solve the problem. As with all shortcuts on the GMAT, if they “click” for you, they’ll save you time, which then you can use for those problems that simply require more thought. Please don’t spend all of your time memorizing tricks, as the GMAT is written to reward “higher order thinking”. That said, a time-saving trick or two can provide you with an additional few minutes on the exam, and the corresponding confidence that you’re primed to post a high score.

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One isn't a prime factor… Therefore there are 7 and not 8 prime factors of 24.

You're right that 1 is not a prime factor, but the question was "How many unique factors does (number) have?" For 24, those are: 1, 2, 3, 4, 6, 8, 12, and 24. The unique PRIME factors of 24 are 2 and 3. Start with the above list, and just pull out the prime numbers.Be careful… On the GMAT, make sure you know exactly what they're asking!

Is there a difference between unique factors of x and factors of x?

similarly with prime factors and unique prime factors?

Would this trick apply for a square? for example 64 = (2^6)(1^1)

(6+1) x(1+1) = 14

or

(6+1) x (1) = 7.

However, the factors are 64 X 1, 32 X 2, 16 x 4, 8 x 8.

This equals 8 factors.